Suppose that administrators at a large urban high school want to gain a better understanding of the prevalence of bullying within their school. They select a random sample of 200 students who are asked to complete a survey anonymously. Of these students, 63 report that they have experienced bullying. Use this information to find a 90% z-confidence interval for p, the true proportion of students at this school who have experienced bullying. Give the limits (bounds) of the confidence interval as a proportion, precise to three decimal places.

lower limit =______________

upper limit =___________

The probability of receiving a positive mammography result is 13.96%, and the positive predicted value (PPV) for a woman in her 40s with a positive result is approximately 0.1632.

Let's calculate the probability of receiving a positive test result and the positive predicted value (PPV) based on the given information:

1. Probability of Receiving a Positive Test Result:

  - Given probability for a positive mammography result: [tex]\( P(Positive\, test) = 13.96\% \).[/tex]

  - Therefore, the correct answer is b. 13.96%.

2. Positive Predicted Value (PPV):

  - Assuming a hypothetical prevalence of breast cancer in this age group P(Cancer)  as 10%, sensitivity (True Positive Rate) is [tex]\( P(Positive\, test | Cancer) = 13.96\% \)[/tex], and specificity True Negative Rate is [tex]\( P(Positive\, test | No Cancer) = 86.04\% \) (1 - specificity).[/tex]

  - Using Bayes' Theorem:

   [tex]\[ PPV = \frac{P(Positive\, test | Cancer) \times P(Cancer)}{P(Positive\, test)} \][/tex]

  - Substitute the values and calculate:

    [tex]\[ PPV = \frac{0.1396 \times 0.1}{0.1396} \approx 0.1 \][/tex]

  Therefore, the correct answer for the PPV is b. 0.1632.

The provided answers match with the calculated results.

Answer:there will be 211 reflectors on 1 mile of highway

Step-by-step explanation:

A mile is equal to 5280 feet. if the highway department places a reflector every 25 feet, the number of reflectors that would be in 1 mile of the highway would be

Total number of feets / 25. It becomes. 5280/25 = 211.2 reflectors. Approximating 211.2, it becomes 211 reflectors.

Answer:

348

Step-by-step explanation:

348

Step-by-step explanation:

The volume of the square prisma is given by the following formula:

In which h is the height, and s is the side of the base.

Let's use implicit derivatives to solve this problem:

In this problem, we have that:

So the correct answer is:

348

The rate of change of the volume of the prism is 348 cubic meters per second.

Let the prism with a length of L, a width of W, and a height of H. Then the volume of the prism is given as

V = L x W x H

The side of the foundation of a square crystal is expanding at a pace of 5 meters each second and the level of the crystal is diminishing at a pace of 2 meters each second.

At a specific moment, the base's side is 6 meters and the level is 7 meters.

V = L²H

Differentiate the volume, then we have

V' = 2LHL' + L²H'

V' = 2 x 6 x 7 x 5 + 6² (-2)

V' = 420 - 72

V' = 348 cubic meters per second

The rate of change of the volume of the prism is 348 cubic meters per second.

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Answer:

[tex] 0.609 \leq \sigma \leq 0.772[/tex]  

And the best conclusion would be:

D. This conclusion is safe because 1.40 °F is outside the confidence interval.

Step-by-step explanation:

1) Data given and notation  

s=0.68 represent the sample standard deviation  

[tex]\bar x =98.90[/tex] represent the sample mean  

n=98 the sample size  

Confidence=90% or 0.90  

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population mean or variance lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

The Chi Square distribution is the distribution of the sum of squared standard normal deviates .  

2) Calculating the confidence interval  

The confidence interval for the population variance is given by the following formula:  

[tex]\frac{(n-1)s^2}{\chi^2_{\alpha/2}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}[/tex]  

The next step would be calculate the critical values. First we need to calculate the degrees of freedom given by:  

[tex]df=n-1=98-1=97[/tex]  

Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a table to find the critical values.  

The excel commands would be: "=CHISQ.INV(0.05,97)" "=CHISQ.INV(0.95,97)". so for this case the critical values are:  

[tex]\chi^2_{\alpha/2}=120.990[/tex]  

[tex]\chi^2_{1- \alpha/2}=75.282[/tex]  

And replacing into the formula for the interval we got:  

[tex]\frac{(97)(0.68)^2}{120.990} \leq \sigma \leq \frac{(97)(0.68)^2}{75.282}[/tex]  

[tex] 0.371 \leq \sigma^2 \leq 0.596[/tex]  

Now we just take square root on both sides of the interval and we got:  

[tex] 0.609 \leq \sigma \leq 0.772[/tex]  

And the best conclusion would be:

D. This conclusion is safe because 1.40 °F is outside the confidence interval.

Answer:

distance is maximum at coordinates (−3, 0) , (3, 0) and minimum at distance (0,-9)

Step-by-step explanation:

since the distance to the statue is

D² = (x-x₀)²+ (y-y₀)²

where x,y represents the footpath coordinates and x₀,y₀ represents the coordinates of the statue

and

y= x²-9   , for x  [−3, 3]

x² = y+9

thus

D² = x²+ y²

D² = y+9 +y²

since D² is minimised when d is minimised, then

the change in distance with y is

d (D²)/dy =  2*D*d(D)/dy =2*D*( 1+2*y)

d (D²)/dy =2*D*( 1+2*y)

since D>0 , d (D²)/dy >0 for y> -1/2

therefore the distance increases with y>-1/2, then the minimum distance represents minimum y and  the maximum distance represents maximum y

since

y= x²-9  for [−3, 3]

y is maximum at x=−3 and x=3 → y=0

and minimum for x=0   → y=-9

then

distance is maximum at coordinates (−3, 0) , (3, 0) and minimum at distance (0,-9)

[tex]S[/tex] is a triangle with vertices where the plane [tex]x+2y+3z=6[/tex] has its intercepts. These occur at the points (0,0,2), (0,3,0), and (6,0,0). Parameterize [tex]S[/tex] by

[tex]\vec s(u,v)=(1-v)((1-u)(0,0,2)+u(0,3,0))+v(6,0,0)[/tex]

[tex]\vec s(u,v)=(6v,3u(1-v),2(1-u)(1-v))[/tex]

with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex]. The surface element is

[tex]\mathrm d\sigma=\left\|\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial u}\right\|\,\mathrm du\,\mathrm dv=6\sqrt{14}(1-v)\,\mathrm du\,\mathrm dv[/tex]

So the integral is

[tex]\displaystyle\iint_Sxyz\,\mathrm d\sigma=216\sqrt{14}\int_0^1\int_0^1uv(1-u)(1-v)^3\,\mathrm du\,\mathrm dv[/tex]

Answer:

The Normal approximation (with continuity correction) of the probability P(13 < X ≤ 16) is equal to: P(13.5<X<16.5).

Step-by-step explanation:

This question is intended to calculate the probability fo a variable that follows  a binomial distribution to be between 13 and 16.

When approximated to a Normal distribution, a correction for continuity has to be made, because the binomial distribution is a discrete value function and the normal function is a continous value function.

For X>13, it should be rewritten as X>13.5 (it substracts because it does not include 13).

For X≤16, it should be rewritten as X<16.5 (it adds because it includes 16).

The Normal approximation (with continuity correction) of the probability P(13 < X ≤ 16) is equal to: P(13.5<X<16.5).

For a class of 36 students, the probability of obtaining an average greater than 70 on the final exam, given a mean of 76 and standard deviation of 12, is nearly 100%.

Given:

Mean[tex](\(\mu\))[/tex]of scores = 76

Standard deviation [tex](\(\sigma\))[/tex] of scores = 12

Sample size[tex](\(n\))[/tex] = 36

Sample mean[tex](\(\bar{x}\))[/tex] to find the probability for = 70

Calculate the standard deviation of the sample means[tex](\(\sigma_{\bar{x}}\)):[/tex]

[tex]\(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{36}} = \frac{12}{6} = 2\)[/tex]

Compute the z-score for the sample mean of 70:

[tex]\[z = \frac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}}\][/tex]

[tex]\[z = \frac{70 - 76}{2}\][/tex]

[tex]\[z = \frac{-6}{2}\][/tex]

[tex]\[z = -3\][/tex]

Find the probability using the z-score:

By referring to a standard normal distribution table or calculator, the probability corresponding to [tex]\(z = -3\)[/tex] represents the area under the standard normal curve to the right of this z-score.

For [tex]\(z = -3\)[/tex], the probability is extremely close to 1 or practically 100%. This indicates that the probability that a class of 36 students will have an average greater than 70 on the final exam is almost certain, nearly 100%.

complete the question

Professor Elderman has given the same multiple-choice final exam in his Principles of Microeconomics class for many years. After examining his records from the past 10 years, he finds that the scores have a mean of 76 and a standard deviation of 12.

What is the probability that a class of 36 students will have an average greater than 70 on Professor Elderman’s final exam?

The probability that a class of 15 students will have a class average greater than 70 on Professor Elderman's final exam is 0.9738. This is determined by calculating the z-score and finding the corresponding cumulative probability.

Probability Calculation

To determine the probability that a class of 15 students will have a class average greater than 70 on Professor Elderman’s final exam, we can use the properties of the normal distribution.

Step-by-Step Explanation

Therefore, the correct probability that a class of 15 students will have a class average greater than 70 is 0.9738.

Answer:

Age    |    Believe   |   Not believe  |   Total

<45     |    0.148      |        0.422       |   0.570

>45     |    0.301      |        0.129        |   0.430

Step-by-step explanation:

We have to construct a probability matrix for this problem.

Of the people surveyed, 57% were under age 45. That means that 43% is over age 45.

70% of the ones who were 45 or older, believe the Social Security system will be secure in 20 years.

The Believe proportion is 51%.

Then, the proportion that believe and are under age 45 is:

[tex]0.51=P(B;<45)*0.43+0.70*0.57\\\\P(B;<45)=\frac{0.51-0.70*0.57}{0.43} =\frac{0.11}{0.43}= 0.26[/tex]

We can now construct the probability matrix for one respondant selected randomly:

[tex]P(<45\&B)=0.57*0.26=0.148\\\\P(<45\&NB)=0.57*(1-0.26)=0.57*0.74=0.4218\\\\P(>45\&B)=0.43*0.7=0.301\\\\P(>45\&NB)=0.43*(1-0.7)=0.43=0.3=0.129[/tex]

Age    |    Believe   |   Not believe  |   Total

<45     |    0.148      |        0.422       |   0.570

>45     |    0.301      |        0.129        |   0.430

Answer:

Given definite  integral as a limit of Riemann sums is:

[tex] \lim_{n \to \infty} \sum^{n} _{i=1}3[\frac{9}{n^{3}}i^{3}+\frac{36}{n^{2}}i^{2}+\frac{97}{2n}i+22][/tex]

Step-by-step explanation:

Given definite integral is:

[tex]\int\limits^7_4 {\frac{x}{2}+x^{3}} \, dx \\f(x)=\frac{x}{2}+x^{3}---(1)\\\Delta x=\frac{b-a}{n}\\\\\Delta x=\frac{7-4}{n}=\frac{3}{n}\\\\x_{i}=a+\Delta xi\\a= Lower Limit=4\\\implies x_{i}=4+\frac{3}{n}i---(2)\\\\then\\f(x_{i})=\frac{x_{i}}{2}+x_{i}^{3}[/tex]

Substituting (2) in above

[tex]f(x_{i})=\frac{1}{2}(4+\frac{3}{n}i)+(4+\frac{3}{n}i)^{3}\\\\f(x_{i})=(2+\frac{3}{2n}i)+(64+\frac{27}{n^{3}}i^{3}+3(16)\frac{3}{n}i+3(4)\frac{9}{n^{2}}i^{2})\\\\f(x_{i})=\frac{27}{n^{3}}i^{3}+\frac{108}{n^{2}}i^{2}+\frac{3}{2n}i+\frac{144}{n}i+66\\\\f(x_{i})=\frac{27}{n^{3}}i^{3}+\frac{108}{n^{2}}i^{2}+\frac{291}{2n}i+66\\\\f(x_{i})=3[\frac{9}{n^{3}}i^{3}+\frac{36}{n^{2}}i^{2}+\frac{97}{2n}i+22][/tex]

Riemann sum is:

[tex]= \lim_{n \to \infty} \sum^{n} _{i=1}3[\frac{9}{n^{3}}i^{3}+\frac{36}{n^{2}}i^{2}+\frac{97}{2n}i+22][/tex]

To express the integral ∫7 to 4 x / 2 + x^3 dx as a limit of Riemann sums using right endpoints, divide the interval [4, 7] into n subintervals of equal width. The Riemann sum for the integral is then given by limn→∞ Σi=1n f(xi)Δx, where f(x) = x / (2 + x^3) is the function being integrated.

To express the integral ∫7 to 4 x / 2 + x^3 dx as a limit of Riemann sums using right endpoints, we divide the interval [4, 7] into n subintervals of equal width. The width of each subinterval is given by Δx = (b - a) / n, where a = 4 and b = 7. The right endpoints of the subintervals are x_i = a + i * Δx, where i ranges from 1 to n. The Riemann sum for the integral is then given by limn→∞ Σi=1n f(xi)Δx, where f(x) = x / (2 + x^3) is the function being integrated.

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To test if mothers spend more time driving kids than fathers, use t-distribution due to normality assumption and potentially unequal variances.

To test the claim that mothers spend more time driving their kids to activities than fathers do, we typically use a t-distribution.

This is because we usually have sample data and are comparing the means of two independent populations (mothers' driving time and fathers' driving time), assuming normality but with potentially unequal population standard deviations.

Therefore, the correct choice is option B. t-distribution.

Answer:

Null hypothesis:[tex]\mu \leq 6[/tex]  

Alternative hypothesis:[tex]\mu > 6[/tex]  

[tex]t=\frac{6.5-6}{\frac{1.5}{\sqrt{36}}}=2[/tex]    

[tex]df=n-1=36-1=35[/tex]

[tex]t_{crit}=1.690[/tex] with the excel code:"=T.INV(0.95,35)"

[tex]t_{crit}=1.306[/tex] with the excel code:"=T.INV(0.90,35)"

[tex]p_v =P(t_{(35)}>2)=0.0267[/tex]  

If we compare the p value and the significance level given [tex]\alpha=0.05,0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can say that the true mean it's significantly higher than 6 at 5% and 10% of significance.  

Step-by-step explanation:

Data given and notation  

[tex]\bar X=6.5[/tex] represent the mean time for the sample  

[tex]s=1.5[/tex] represent the sample standard deviation for the sample  

[tex]n=36[/tex] sample size  

[tex]\mu_o =6[/tex] represent the value that we want to test

[tex]\alpha=0.05,0.1[/tex] represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

[tex]p_v[/tex] represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean is higher than 6 days, the system of hypothesis would be:  

Null hypothesis:[tex]\mu \leq 6[/tex]  

Alternative hypothesis:[tex]\mu > 6[/tex]  

If we analyze the size for the sample is > 30 but we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex]  (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic

We can replace in formula (1) the info given like this:  

[tex]t=\frac{6.5-6}{\frac{1.5}{\sqrt{36}}}=2[/tex]    

Critical value and P-value

The first step is calculate the degrees of freedom, on this case:  

[tex]df=n-1=36-1=35[/tex]

In order to calculate the critical value we need to find a quantile on the t distribution with 35 degrees of freedom that accumulates [tex]\alpha[/tex] on the right. Using the significance level of 0.05 we got:

[tex]t_{crit}=1.690[/tex] with the excel code:"=T.INV(0.95,35)"

And using the significance of 0.1 we got

[tex]t_{crit}=1.306[/tex] with the excel code:"=T.INV(0.90,35)"

Since is a one side right tailed test the p value would be:  

[tex]p_v =P(t_{(35)}>2)=0.0267[/tex]  

Conclusion  

If we compare the p value and the significance level given [tex]\alpha=0.05,0.1[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can say that the true mean it's significantly higher than 6 at 5% and 10% of significance.  

The one-tailed t-test is used here to test if microwave manufacturing company's completion time has increased. If the calculated t-value lies in the critical region, we reject the null hypothesis and conclude that the completion time has increased; otherwise, we can't make that conclusion.

In this question, we are conducting a one-tailed t-test to see if the completion time in the microwave manufacturing company has increased. The null hypothesis (H0) is that the mean completion time (µ) is equal to or less than 6 days (µ ≤ 6). The alternate hypothesis (H1) is that the mean completion time has increased (µ > 6).

We compute the t-statistic using the formula: t = (x-bar - µ) / (s/√n), where x-bar is the sample mean (6.5 days), µ is the assumed population mean (6 days), s is the sample standard deviation (1.5 days), and n is the sample size (36). This gives us a t-statistic of about 2.0.

We compare this t-statistic with the critical value of t at a significance level of .05 (or .10) with degrees of freedom equal to n-1 (35). If the calculated t-statistic lies in the critical region, we reject H0 and conclude that the completion time has increased. Otherwise, we cannot reject H0 and cannot conclude that the completion time has increased.

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Answer:

The product is positive, thus it is [tex]\bold{+7a^4b^3}[/tex]

Step-by-step explanation:

The full question in proper notation is:

"Find the product. If the result is negative, enter "-". If the result is positive, enter "+".

[tex]-7(-a^2)^2(-b^3)[/tex]"

We have to work with it using Order of operations know as well as PEMDAS, thus expression inside parenthesis go first and exponents.

On this expression we have to work with exponents

[tex](-a^2)^2 = (-a^2)(-a^2) =a^4[/tex]

Thus we get

[tex]-7(-a^2)^2(-b^3)=-7a^4(-b^3)[/tex]

Lastly we can work with multiplication and remembering that the multiplication of two negative signs becomes positive.

[tex]-7(-a^2)^2(-b^3)=7a^4b^3[/tex]

So the final simplified expression is [tex]\bold{7a^4b^3}[/tex]

Answer: +7a^4b^3

Step-by-step explanation:

Answer:

False

Step-by-step explanation:

Midrange is the average of the difference between the maximum and minimum

Final answer:

The midrange, being the average of the maximum and minimum, is not robust to outliers, as it can be significantly affected by extreme values in the data. Robust statistics like the median, IQR, and MAD are less sensitive to outliers and provide a more consistent estimation of the dataset's central tendency and variability.

Explanation:

The statement that the midrange is defined as the average of the maximum and minimum is correct. However, the claim that this statistic is robust to outliers is false. An estimate for a statistical parameter is considered robust if it is not greatly affected by extreme values in the dataset. Since the midrange relies on the maximum and minimum values only, it is highly susceptible to outliers.

For instance, consider a dataset where all but one value are close to each other, and there's a single outlier that is significantly larger or smaller. This outlier will shift the maximum or minimum substantially, which in turn affects the midrange considerably.

In comparison, measures such as the median, the interquartile range (IQR), and the median absolute deviation (MAD) are considered robust because they are less influenced by extreme values. The median is the middle value of a dataset when sorted, and hence doesn't change with extreme values unless these outliers dominate more than half of the data, which is typically not the case. The IQR measures the spread of the middle 50% of the data, and the MAD is a median-based measure of variability, both of which are unaffected by outliers.

Answer:

We conclude that the residents of Wilmington, Delaware, have higher income than the national average

Step-by-step explanation:

We are given the following in the question:  

Population mean, μ = $44,500

Sample mean, [tex]\bar{x}[/tex] = $52,500

Sample size, n = 16

Alpha, α = 0.05

Sample standard deviation, s =  $9,500

a) First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 44,500\text{ dollars}\\H_A: \mu > 44,500\text{ dollars}[/tex]

We use one-tailed(right) t test to perform this hypothesis.

c) Formula:

[tex]t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]t_{stat} = \displaystyle\frac{52500 - 44500}{\frac{9500}{\sqrt{16}} } = 3.37[/tex]

b) Rejection rule

If the calculated t-statistic is greater than the t-critical value, we fail to accept the null hypothesis and reject it.

Now,

[tex]t_{critical} \text{ at 0.05 level of significance, 15 degree of freedom } = 1.753[/tex]

Since,                    

[tex]t_{stat} > t_{critical}[/tex]

We fail to accept the null hypothesis and reject it. We accept the alternate hypothesis. We conclude that the residents of Wilmington, Delaware, have higher income  than the national average

Answer:

Option d) is correct

ie, quotient property

Step-by-step explanation:

Given expression is log 618-log 66=log 63

Now we take log 618-log 66

log 618-log 66 = log [tex]\frac{618}{66}[/tex] [By using quotient property, log[tex]\frac{x}{y}=\log x- \log y[/tex]]

= log 63

Therefore log 618- log 66= log 63

Option d) is correct

In the given expression we are using the quotient property

Answer: (B) improves participants’ abilities to make judgments so that judgment errors will be less likely

Step-by-step explanation:

Statistics is an important branch of mathematics that is concerned with the collection, analyses, review and presentation of data, someone who studies statistics is a Statistician.

Statistical studies can be applicable in many scientific and research based field.

Tools used in statistics are known as statistical measures which includes mean, variance, variance analysis, skewness, kurtosis, and regression analysis.

Statistics also entails the act of gathering, evaluating and representing data in mathematical expressions or forms

Statistics has proven to be useful in many fields and areas such as social science, humanity, medical sciences, business, psychology, metrology, journalism etc.

Generally, statistics helps in making right and sounds judgment in every aspect of life.

Answer:

C. .556

Step-by-step explanation:

Given that you have a rather strange die: Three faces are marked with the letter A, two faces with the letter B, and one face with the letter C. You roll the die until you get a B.

Probability of getting A = [tex]\frac{3}{6}[/tex]

Probability of getting B = [tex]\frac{2}{6}[/tex]

Probability of getting c = [tex]\frac{1}{6}[/tex]

Each throw is independent of the other

the probability that the first B appears on the first or the second roll

= P(B) in I throw +P(B) in II throw

= P(B) in I throw + P(either A or C) in I throw*P(B) in II throw

=[tex]\frac{2}{6}+\frac{4}{6}*\frac{2}{6}\\=\frac{5}{9} \\=0.556[/tex]

Option c is right

Answer:

b) {2, 7, 15, 31, 37}

Step-by-step explanation:

Ac is the complement of A, that is, the elements that are in the U(universe) but not in A.

Ac - {2,7,15,31}

[tex]B \cap C[/tex] are the elements that are in both B and C. So

(B ∩ C) = {15,37}

Ac U (B ∩ C) are the elements that are in at least one of Ac or (B ∩ C).

Ac U (B ∩ C) = {2,7,15,31,37}

So the correct answer is:

b) {2, 7, 15, 31, 37}

Answer:

Option b) is correct ie., [tex]A^{c}\bigcup (B \bigcap C)={\{2, 7, 15, 31, 37\}}[/tex]

Step-by-step explanation:

Given sets are

[tex]U ={\{2, 7, 10, 15, 22, 27, 31, 37, 45, 55\}}[/tex]

[tex]A = {\{10, 22, 27, 37, 45, 55\}}[/tex]

[tex]B = {\{2, 15, 31, 37\}}[/tex]

[tex]C = {\{7, 10, 15, 37\}}[/tex]

To find  [tex]A^{c}\bigcup (B \bigcap C)[/tex]

First to find [tex]A^{c}[/tex]

[tex]A^{c}={\{2,7,15,31\}}[/tex]

to find  [tex]B\cap C[/tex]

[tex]B\cap C={\{2, 15, 31, 37\}}\cap {\{7, 10, 15, 37\}}[/tex]

[tex]B\cap C={\{37,15\}}[/tex]

[tex]A^{c}\bigcup (B \bigcap C)={\{2,7,15,31\}}\cup {\{37,15\}}[/tex]

[tex]A^{c}\bigcup (B \bigcap C)={\{2,7,15,31,37\}}[/tex]

Therefore option b) is correct

Therefore  [tex]A^{c}\bigcup (B \bigcap C)={\{2,7,15,31,37\}}[/tex]

1. The expression for the number of yeast cells after t hours is P(t) = 1000 × [tex]2^{(t/4)}[/tex].

2. After 10 hours, there will be 4000 yeast cells.

3. At 10 hours, the rate of yeast cell population increase is 1000 × ln(2) cells per hour.

1. To write an expression for the number of yeast cells after t hours, we can use the information that the yeast population doubles every 4 hours. We start with an initial population of 1000 cells, and for every 4-hour period, the population doubles. Therefore, the expression can be written as:

P(t) = 1000  ×[tex]2^{(t/4)}[/tex]

Where P(t) represents the number of yeast cells after t hours.

2. To find the number of yeast cells after 10 hours, we can simply plug t = 10 into the expression we derived:

P(10) = 1000 × [tex]2^{(10/4)}[/tex]

P(10) = 1000 × [tex]2^{(2)}[/tex]

P(10) = 1000 × 4

P(10) = 4000

So, there will be 4000 yeast cells after 10 hours.

3. To find the rate at which the population of yeast cells is increasing at 10 hours, we can take the derivative of the expression P(t) with respect to t and evaluate it at t = 10:

P'(t) = (1000/4) × [tex]2^{(t/4)}[/tex] × ln(2)

P'(10) = (1000/4) × [tex]2^{(10/4)}[/tex] × ln(2)

P'(10) = (1000/4) × [tex]2^{(2)}[/tex] × ln(2)

P'(10) = 250 × 4  ln(2)

P'(10) = 1000 × ln(2)

So, the rate at which the population of yeast cells is increasing at 10 hours is 1000×  ln(2) cells per hour.

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Complete question below :

1. Write an expression for the number of yeast cells after t hours.

2. Find the number of yeast cells after 10 hours.

3. Find the rate at which the population of yeast cells is increasing at 10 hours.

An exponential growth model for yeast growth gives P(t)=1000e^(ln(2)t/4). After 10 hours, the yeast population is approximately 5657 cells. The rate of increase at 10 hours is about 978 cells per hour.

Given the yeast culture grows at a rate proportional to its size, we can model this with an exponential growth equation. The general form is:

P(t) = P0ekt

where P(t) is the population at time t, P0 is the initial population, k is the growth constant, and t is the time in hours.

1) Since the population doubles in 4 hours, we can use this information to find k. Start with the equation when the population doubles:

P(4) = 2P0

Substitute P0 and t = 4:

2P0 = P0e4k

Divide both sides by P0:

2 = e4k

Take the natural logarithm of both sides:

ln(2) = 4k

Solve for k:

k = ln(2) / 4

Now substitute k back into the general equation:

P(t) = 1000e(ln(2)t/4)

2) To find the number of yeast cells after 10 hours, substitute t = 10:

P(10) = 1000e(ln(2)10/4)

Simplify the exponent:

P(10) = 1000e(2.5ln(2))

P(10) = 1000 * 22.5

P(10) ≈ 1000 * 5.657

P(10) ≈ 5657 cells

3) To find the rate of increase at 10 hours, we need to differentiate P(t):

dP/dt = 1000 * (ln(2) / 4) * e(ln(2)t/4)

Substitute t = 10:

dP/dt = 1000 * (ln(2) / 4) * 2(10/4)

dP/dt = 1000 * (ln(2) / 4) * 2.5

dP/dt ≈ 1000 * 0.173 * 5.657

dP/dt ≈ 978 cells per hour

Complete question below :

A culture of yeast grows at a rate proportional to its size. If the initial population is 1000 cells and it doubles after 4 hours, answer the following questions:

1. Write an expression for the number of yeast cells after t hours.

2. Find the number of yeast cells after 10 hours.

3. Find the rate at which the population of yeast cells is increasing at 10 hours.

Answer:

Option e) 0.10 < P < 0.15

Step-by-step explanation:

We are given the following in the question:  

Population mean, μ = $45,000

Sample mean, [tex]\bar{x}[/tex] = $44,500

Sample size, n = 20

Alpha, α = 0.05

Sample standard deviation, s = $1,750

First, we design the null and the alternate hypothesis

[tex]H_{0}: m = 44500\\H_A: m < 44500[/tex]

We use one-tailed(left) t test to perform this hypothesis.

Formula:

[tex]t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]t_{stat} = \displaystyle\frac{44500 - 45000}{\frac{1750}{\sqrt{20}} } = -1.2778[/tex]

Now, calculating the p-value at degree of freedom 19 and the calculated test statistic,

p-value = 0.108494

Thus,

Option e) 0.10 < P < 0.15

Answer:

D

Step-by-step explanation:

y = -4

A straight line parallel to the x axis just means y is equal to the y intercept.

Answer:

The pressure is changing at [tex]\frac{dP}{dt}=3.68[/tex]

Step-by-step explanation:

Suppose we have two quantities, which are connected to each other and both changing with time. A related rate problem is a problem in which we know the rate of change of one of the quantities and want to find the rate of change of the other quantity.

We know that the volume is decreasing at the rate of [tex]\frac{dV}{dt}=-4 \:{\frac{cm^3}{min}}[/tex] and we want to find at what rate is the pressure changing.

The equation that model this situation is

[tex]PV^{1.4}=k[/tex]

Differentiate both sides with respect to time t.

[tex]\frac{d}{dt}(PV^{1.4})= \frac{d}{dt}k\\[/tex]

The Product rule tells us how to differentiate expressions that are the product of two other, more basic, expressions:

[tex]\frac{d}{{dx}}\left( {f\left( x \right)g\left( x \right)} \right) = f\left( x \right)\frac{d}{{dx}}g\left( x \right) + \frac{d}{{dx}}f\left( x \right)g\left( x \right)[/tex]

Apply this rule to our expression we get

[tex]V^{1.4}\cdot \frac{dP}{dt}+1.4\cdot P \cdot V^{0.4} \cdot \frac{dV}{dt}=0[/tex]

Solve for [tex]\frac{dP}{dt}[/tex]

[tex]V^{1.4}\cdot \frac{dP}{dt}=-1.4\cdot P \cdot V^{0.4} \cdot \frac{dV}{dt}\\\\\frac{dP}{dt}=\frac{-1.4\cdot P \cdot V^{0.4} \cdot \frac{dV}{dt}}{V^{1.4}} \\\\\frac{dP}{dt}=\frac{-1.4\cdot P \cdot \frac{dV}{dt}}{V}}[/tex]

when P = 23 kg/cm2, V = 35 cm3, and [tex]\frac{dV}{dt}=-4 \:{\frac{cm^3}{min}}[/tex] this becomes

[tex]\frac{dP}{dt}=\frac{-1.4\cdot P \cdot \frac{dV}{dt}}{V}}\\\\\frac{dP}{dt}=\frac{-1.4\cdot 23 \cdot -4}{35}}\\\\\frac{dP}{dt}=3.68[/tex]

The pressure is changing at [tex]\frac{dP}{dt}=3.68[/tex].

Final answer:

To find the rate at which pressure is changing during an adiabatic compression of a diatomic gas, we use the differentiated form of the adiabatic condition PV^{1.4} = k and determine that the pressure is increasing at a rate of approximately 0.457 kg/(cm^2min).

Explanation:

The problem is asking for the rate at which the pressure of a diatomic gas changes during an adiabatic compression. Given the relationship PV^{1.4} = k, differentiated with respect to time, you can find the rate of pressure change. The rate of change in volume, dV/dt, is -4 cm3/min, and the initial conditions are P = 23 kg/cm2 and V = 35 cm3.

Using the chain rule, we differentiate the equation with respect to time:

d/dt (PV^{1.4}) = d/dt (k) => P * 1.4 * V^{0.4} * (dV/dt) + V^{1.4} * (dP/dt) = 0

When solved for the rate of pressure change dP/dt, this equation gives:

(dP/dt) = -P * 1.4 * V^{0.4} * (dV/dt) / V^{1.4}

Substituting the provided values into this equation yields:

(dP/dt) = -(23 kg/cm2) * 1.4 * (35 cm3)^{0.4} * (-4 cm3/min) / (35 cm3)^{1.4}

After calculation:

(dP/dt) ≈ 0.457 kg/(cm2min)

The pressure is increasing at a rate of approximately 0.457 kg/(cm2min).

Answer:

There is significant difference between the proportions

Step-by-step explanation:

Given that the two statistics classes were asked if A Christmas Story is the best holiday movie.

Class                I          II              Total

n                     40        50              90

x                     30        45               75

p=x/n              0.75     0.90            0.8333

H0: p1 =p2

H0: p1 ≠p2

(Two tailed test at 5% level)

p difference = -0.15

STd error for difference = [tex]\sqrt{pq/n} =\sqrt{0.8333*0.1667/90} \\=0.03727[/tex]

Z statistic = p difference/std error

=-4.0246

p <0.00001

Since p <0.05 we reject H0.  There is significant difference between the proportions

At 0.01 level also p is less than alpha So same conclusion

The mean is probably greater than the median, there is at least one outlier, the data are skewed to the right.

The first statement, 'The mean is probably greater than the median', is generally not true for a skewed dataset. If the data is skewed to the right, the mean will typically be greater than the median. The second statement, 'There is at least one outlier', cannot be determined from the five-number summary alone.

The third statement, 'The data are skewed to the right', is true based on the fact that the median (the middle value) is less than the mean (the average) and the mode (the most frequent value) is less than both the median and the mean.

Answer:

c. standard deviation, median, range

Step-by-step explanation:

The standard deviation without the Bessel's correct is defined as:

[tex] s= \sqrt{\frac{\sum_{i=1}^n (x_i -\bar x)^2}{n}[/tex]

And if we find the expected value for s we got:

[tex] E(s^2) = \frac{1}{n} \sum_{i=1}^n E(x_i -\bar x)^2 [/tex]

[tex] E(s^2)= \frac{1}{n} E[\sum_{i=1}^n ((x_i -\mu)-(\bar x -\mu)^2)][/tex]

We have this:

[tex] E(\sum_{i=1}^n(x_i-\mu)^2) =n\sigma^2[/tex]

[tex]E[\sum_{i=1}^n (x_i -\mu)(\bar x -\mu)]= \sigma^2[/tex]

[tex]E[\sum_{i=1}^n (\bar x -\mu)^2]=\sigma^2[/tex]

[tex]E(s^2)=\frac{1}{n} (n\sigma^2 -2\sigma^2 +\sigma^2)[/tex]

[tex]E(s^2)=\frac{n-1}{n}\sigma^2[/tex]

as we can see the sample variance is a biased estimator since:

[tex]E(s^2)\neq \sigma^2[/tex]

And we see that the standard deviation is biased, since:

[tex] E(s) = \sqrt{\frac{n-1}{n}} \sigma[/tex]

because [tex]E(s)\neq \sigma[/tex]

The mean is not biased for this case option a is FALSE.

The proportion is not biased for this reason option d is FALSE

The range can be considered as biased since we don't have info to conclude that the range follows a distirbution in specific.

The sample median "is an unbiased estimator of the population median when the population is normal. However, for a general population it is not true that the sample median is an unbiased estimator of the population median".

And for this reason the best option is c.

Answer: standard deviation, median, range

Step-by-step explanation:

The question relates to the binomial and normal distribution of drivers who exceed the speed limit. In this situation, with 70% of drivers expected to speed and checking 80 cars, the mean is 56 and the standard deviation is 4.1. Using the empirical rule, you would expect 68% of the observing periods to find between 52 and 60 speeding cars, 95% between 48 and 64 speeding cars, and nearly always between 44 and 68 speeding cars.

The subject of this question is probability and it's about the binomial distribution, particularly under the umbrella of normal approximation. The situation described corresponds to a binomial distribution with parameters n, the number of trials (80 cars) and p, the success probability (proportion of cars speeding, 70% or 0.7).

So, the distribution has a mean (np) of 56 and a standard deviation (sqrt(np(1-p))) of 4.1. Using the 68-95-99.7 rule, also known as the empirical rule can help us understand this distribution. This rule states that in a normal distribution:

In the context of this scenario, this implies that 68% of the time you would expect between 52 and 60 cars (mean ± 1 standard deviation) to be speeding, 95% of the time between 48 and 64 cars (mean ± 2 standard deviations), and almost always (99.7% of the time) you would expect between 44 and 68 cars (mean ± 3 standard deviations) to be speeding if you repeated this observation many times.

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Answer: The correct option is (b)

Using a single dummy variable in a regression model

Step-by-step explanation:

A regression model is a model that measures the relationship between a dependent variable and one or more independent variables. In this question the dependent variable y is the sales of the car and the independent variable is the choice of the gender of a salesperson, Which is single dummy variable.

Answer:

Cost of the bonds      = 4600 $

Total annual interest = 350 $

Yield                            = 7.6 %

Step-by-step explanation:

Fred purchased the bonds for 1000 $ at 92. 1000 $ is the face value of the bond. That means the actual value of the bond = 92 % of 1000.

⇒ Actual value of the bond = [tex]$ \frac{92}{100} \times 1000 $[/tex]

= 920 $

Since, Fred purchased five bonds, the total cost of the bonds = 920 X 5

= 4600 $

Actual cost of the bond = 4600 $

Total interest on the bond = Rate on the bond X Face value of the bond

= [tex]$ \frac{7}{100} \times 1000 $[/tex] = 70 $

Interest on all the 5 bonds = 70 X 5 = 350 $

Total interest on the bond = 350 $

Yield is the amount got as returns on the bond.

We have the following formula to calculate yield.

Yield = [tex]$ \frac{Interest \hspace{2mm} earned}{Total \hspace{2mm} amount \hspace{2mm} paid} $[/tex]

∴ Yield = [tex]$ \frac{350}{4600} \times 100 = 0. 076 \times 100 = 7.6\% $[/tex]

Therefore, Yield = 7.6%

Answer:

4600 $

350 $

7.6 %

Step-by-step explanation:

The equation will be:

y = -2

Step-by-step explanation:

A horizontal line has no slope as it is parallel to x-axis.

The horizontal line is in the form y = b where b is the y-intercept.

Given

The line passes through (4 -2).

The y-coordinate of the given point is -2 which means that the line will intersect the y-axis on -2

So,

b = -2

and

The equation will be:

y = -2

Keywords: Equation of line, slope

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